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Question 7: Consider 10 male students Mi, M2,..., Mio and 7 female studertts Fi, F,...,F Assume these 17 students are arranged on a horizontal line such that no two female students are standing next to each other. We are interested in the number of such arrangements, where the order of the students matters 9 4210 Explain what is wrong with the following argument: We are going to use the Product Rule: . Task 1: Arrange the 7 females on a line. There are 7! ways to do this. Task 2: Choose 6 males. There are () ways to do this. females. There are 6! ways to do this. with 4 males that have to be placed . Task 3: Place the 6 males chosen in Task 2 in the 6 gaps between the . Task 4: At this moment, we have arranged 13 students on a line. We are left - Task 4.1: Place one male. There are 14 ways to do this. - Task 4.2: Place one male. There are 15 ways to do this. - Task 4.3: Place one male. There are 16 ways to do this. - Task 4.4: Place one male. There are 17 ways to do this. By the Product Rule, the total number of ways to arrange the 17 students is equal to 7! 6, 14-15-16-17 43, 528,181, 760,000. Question 8: Let n 2 2 be an integer and consider the set S (1,2 . An ordered triple (A, x,y) is called awesome, if (i) AS S, (ii) z e A, and (ii) yE A . Let k be an integer with 1 s k s n. Determine the number of awesome triples (A, r,y) with |Al- k. Prove that the number of awesome triples (A, z,y) with -y is equal to 7-2-1 Determine the number of awesotne triples (A, z,v) with z #x Use the above results to prove that km1
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